# Beyond Pythagoras

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Beyond Pythagoras What this coursework has asked me to do is to investigate and find a generalisation, for a family of Pythagorean triples. This will include odd numbers and even numbers. I am going to investigate a family of right-angled triangles for which all the lengths are positive integers and the shortest is an odd number. I am going to check that the Pythagorean triples (5,12,13) and (7,24,25) cases work; and then spot a connection between the middle and longest sides. The first case of a Pythagorean triple I will look at is: [IMAGE] [IMAGE][IMAGE][IMAGE]The numbers 5, 12 and 13 satisfy the connection. 5² + 12² = 13² 25 + 144 = 169 169 = 13 The second case of a Pythagorean triple I will look at is: [IMAGE][IMAGE]The numbers 7, 24 and 25 satisfy the connection. 7² + 24² = 25² [IMAGE][IMAGE][IMAGE]49 + 576 = 625 625 = 25 There is a connection between the middle and longest side. This is that there is a one number difference. So if M= middle and L= longest L = M + 1 I am going to use the triples, (3,4,5), (5,12,13) and (7,24,25) to find other triples. Then I will put my results in a table and look for a pattern that will occur. I will then try and predict the next results in the table and prove it. [IMAGE] [IMAGE] [IMAGE] n Smallest O Middle O Longest O 1 3 4 5 2 5 12 13 3 7 24 25 There is a clear pattern between the middle and longest side. There is also a sequence forming. n = 1 S = 3 M = 4 L = 5 n = 2 S = 5 M = 12 L = 13